# Training-efficient density quantum machine learning

arXiv: https://arxiv.org/abs/2405.20237 · HTML: https://mlsanity.com/paper/2405.20237/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 112 references that are in the map.
- First posted on arXiv 30 May 2024.

## Builds on (most cited first, 30 of 49)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Scaling Vision with Sparse Mixture of Experts](https://mlsanity.com/paper/2106.05974/) (2021)
- [Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm](https://mlsanity.com/paper/1712.01815/) (2017)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Learning Factored Representations in a Deep Mixture of Experts](https://mlsanity.com/paper/1312.4314/) (2013)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [On the Representation Collapse of Sparse Mixture of Experts](https://mlsanity.com/paper/2204.09179/) (2022)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Distributed Gaussian Processes](https://mlsanity.com/paper/1502.02843/) (2015)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [TensorFlow Quantum: A Software Framework for Quantum Machine Learning](https://mlsanity.com/paper/2003.02989/) (2020)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)

## Built on by (most cited first, 11 of 11)

- [Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks](https://mlsanity.com/paper/2406.18316/) (2024)
- [QuIC: Quantum-Inspired Compound Adapters for Parameter Efficient Fine-Tuning](https://mlsanity.com/paper/2502.06916/) (2025)
- [Minimizing classical resources in variational measurement-based quantum computation for generative modeling](https://mlsanity.com/paper/2604.11578/) (2026)
- [Scalable On-Hardware Training of Quantum Neural Networks and Application to Clinical Data Imputation](https://mlsanity.com/paper/2606.03517/) (2026)
- [Quantum Machine Learning for Industrial Applications](https://mlsanity.com/paper/2606.14822/) (2026)
- [Scalable Message-Passing Quantum Graph Neural Networks in the Weisfeiler-Leman Hierarchy](https://mlsanity.com/paper/2606.26873/) (2026)
- [PERM EQ x GRAPH EQ: Equivariant Neural Networks for Quantum Molecular Learning](https://mlsanity.com/paper/2512.05475/) (2025)
- [Hybrid Quantum-Classical Mixture of Experts: Unlocking Topological Advantage via Interference-Based Routing](https://mlsanity.com/paper/2512.22296/) (2025)
- [Adaptive directional gradients for parameterised quantum circuits](https://mlsanity.com/paper/2606.09734/) (2026)
- [Stacking the Deck: Tunable Trainability in Stacked LCUs](https://mlsanity.com/paper/2607.24686/) (2026)
- [Representational separation between unitary and channel quantum generative models via shared classical randomness at shallow depth](https://mlsanity.com/paper/2608.05110/) (2026)

## Abstract (arXiv)

Quantum machine learning (QML) requires powerful, flexible and efficiently trainable models to be successful in solving challenging problems. We introduce density quantum neural networks, a model family that prepares mixtures of trainable unitaries, with a distributional constraint over coefficients. This framework balances expressivity and efficient trainability, especially on quantum hardware. For expressivity, the Hastings-Campbell Mixing lemma converts benefits from linear combination of unitaries into density models with similar performance guarantees but shallower circuits. For trainability, commuting-generator circuits enable density model construction with efficiently extractable gradients. The framework connects to various facets of QML including post-variational and measurement-based learning. In classical settings, density models naturally integrate the mixture of experts formalism, and offer natural overfitting mitigation. The framework is versatile - we uplift several quantum models into density versions to improve model performance, or trainability, or both. These include Hamming weight-preserving and equivariant models, among others. Extensive numerical experiments validate our findings.